A MODEL FOR TEE EXPERIMEJilTAL MEASUREMEJiIT OF TIlE UTILITY OF GAMBLIJilG

BY HALSEY L,ROYDEN, AND KAROLWJ,.LSH

TECHNICAL REPORT NO. 14 SEPTEMBER 25, 1957

PREPARED UNDER CONTRACT Nonr 225(17) (NR 171-034) FOR OFFICE OF NAVAL RESEARCH

REPRODUCTION IN WHOLE OR IN PART IS PERMITTED FOR ANY PURPOSE OF TEE UlUTED STATES GOVERNMENT

BEHAVIORJ,.LSCIENCES DIVISION

APPLIED MA'l'IlEMATICS AND STATISTICS LABORATORY

STANFORD J CALIFORNI{-I. A MODEL FOR THE EXPERIMENTAL MEASUREMENT OF THE UTILITY OF GAMBLING

by Halsey L. Royden, Patrick Suppes and Karol Walsh

The Model

The experiment described in this paper was designed to measure utility of gambling to an individual and to predict) on the basis of this measured utility, the individual's future choices. The vexing problem of the existence of a specific utility of gambling has been much discussed in the literature of decision making. Von Neumann and Morgenstern ([5], p. 28) have this to say: "We have practically defined numerical utility as being that thing for which the calculus of mathematical expectationS is legitilllate. Since

[our axioms] secure that that necessary construction can be carried out) concepts like a 'specific utility of gambling' cannot be formulated free of cOntradiction on this level." And they add in a footnote to this last sentence: "This may seem to be a para.doxical assertion. But anybody who ha.s seriously tried to axiomatize that elusive concept) will probably concur in it. n As fa.r as we know the work reported in this paperconstitutes the first systematic attempt to measure experimentally the utility of gambling.

The model at the basis of our experiment does not yield a complete axiomatiza­ tion of this "elusive" concept) but it is a mathematically definite model for monetary outcomes.

The model can best be explained by presenting a detailed illustration of how it is tested in the experiment. Offers in the form of a game ma.trix -2-

AB ~ ~

are g:i.ven §, where x and yare d:i.fferent small amounts of money, and z is some other small amount of money approximately half-way between x and y. S is asked to select either option A or option B, and is told that if he chooses option A, he will be allowed to throw a fair die to decide which value he will actually win; if he chooses option B, his win is certain, the amount z. The amount z is varied during the experiment until S's point of indifference between the "sure-thing option" (z,z) and the "gamble"(x,y) is reached, that is, until the value Zl is found such that §, when presented at random the offer

A B EIj D3

shifts back and forth in his choice of A or B, and no longer consistently picks one or the other. The value Zl is called S's indifference point.

This indifference point, z', minus the mean of the "true" option (x,y) is the utility of gambling value associ~ted with the pair of values x and y.

In symbols,

Z 1 - 2;;:x+y tf'l ()x,y -3-

whElre qJ' is the utility of gambling function. Since Elach option (x,y) is uniquely ~EltElrminEld by its mElan m = X;y and its absolutEl diffElrElncEl d = I x-yI, WEl replacEl qJ I by thEl function cp dElfinEld for the mElans !lnd diffElrElncEls, that is,

cp(lx-y I, X~y) = cp'(x,y).

In viElW of the fact that thEl fair diEl USEld is tElstEld to insurEl that it has subjElctivEl probability ~, thEl diffElrElncEl I x-y I is simply twicEl the standard dElviation of the,offElr (x,y), for

2 cr (x,y) whElncEl

1 cr(x,y) = '2 Ix-y I.

ThEl ElxpElrimental problElm is thEln, on the basis of limitEld obsElrvations, to construct each S's utility of gambling surfacEl. As would bEl ElxpElctEld, our procEldurEl is to draw a contour map of cp-valuEls, where means are plotted on the abscissa and diffElrences on the ordinate.

The standard hypothesis of rational bElhavior that individuals maximizEl expected utility is now rElplaced, for the special case of monetary outcomes, by the hypothElsis that -they maximizEl thEl ~ of the expected monetary value and thEl utility of gambling of the option. The latter hypothesis is tested by giving subjects offers of the 'form: -4-

AB c=r:==J..z CLI3 where x, y, z and ware again small amounts of money now satisfying the

condition that x> z > w> y. Once more Ss choose columns knowing that row values will be decided by the throw of a die. It is predicted that

option (x,y) will be chosen over option (z,w) if and only if

y Z+w ~x+y + ~ (I X-y I , ~x+ ) ~ ~ + ~ (I z-w I ,~,z+W)

Experimental procedure

The experimental set-up here reported was the same as that described

in Suppes and Walsh [4]. In fact, the same Ss (eight sailors from the

Moffett J,i'ield Air Base) and an additional eight undergraduate students from

Stanford University (six boys and two gids) were used. When fjrst introduced to the experiment each §. was told that he would be participating in a

"gambling" situation designed to give information about how people make decisions involving various amounts of money. Each S was assured that the experimental results would remain confidential and that these results would not be used to make value judgments about his character or intelligence.

The purpose of the experiment, §.S were told, was purely descr'lptive;

After these preliminary remarks had been made each §., protected by

¢2.00 playing credit, was given a set of trial alternatives of the following sort: -5-

AB CGJ ~ where x, y, z and ware small amounts of money such that x > z > w > y.

S was asked to pick an option, column A or column B, and was told that the row value would be decided by'his tossing an unbiased die. On each side of the die was a nonsense syllable instead of a number (to avoid the effect of previously conceived preference for certain numbers), and S was asked to assign One row value to three sides of the die, the second row value being then automatically assigned to the remaining three sides [I]. Ss were assured that they would not lose more than their ¢2.00 credit during the experiment, and they were told that the average prOfit in previous runs had been about ¢2.50 for the six sessions.~/ In order to save time Ss were not allowed to toss for all offers presented in each session, but at the end of a period of play they were instructed to draw ten numbers at random from an envelope containing as many numbers as there were offers in the session and their wins were the results of these ten tosses.

Offers presented in the first two sessions consisted of various cOmbina- tions .of the seven amounts: -391, -231, -101, 21, 131, 271, 421, such that no option dominated, or was better in both values than its companion option.

~/ The experiment consisted of six sessions. The results of the second two were used to obtain the utility of gambling surfaces as above explained. Data from the first two, for the sailors, were employed in another experi- ment [4] designed to measure monetary utility. The results of both the first and last two sessions for both sets of Ss were used for predic­ tions. -6-

The eight sailors were given 42 offers, containing seven repetitions, and no

"non-gambles", or pairs of alternatives of the form (x,y) and (z,z). The eight students were given 80 offers including all alternatives given to the sailors, plus 3 more repetitions and 35 non-gambles.

Offers yielding the values which were used in constructing the utility of gambling surfaces described earlier were composed of 27 "true" options consisting of various combinations of the 20 amounts: -48i, -38i, -34i,

-31i, -25i, -19i, -lSi, -12i, -6i, Ii, 4i, 12i, 14i, lSi, 21i, 27i, 33i, 34i, 40i, 46i, coupled with sure-thing offers approximately half-way between ti;le two amounts of the "true" option$, and varied until the point of indif­ ference was found, as explained earlier. More specifically, the amount z was found such that the "true" option was chosen over (z, z), but

(z + li, z + li) over the "true" option.

The 70 offers of the last two sessions were composed of pairs of "true" options neither of which dominated the other in both rows. The options cOnsisted of 90 amounts ranging in value from -46i to +50i. Both the first two and last two ses$ions were used for making predictions.

Results

?redictions were made for the data in sessions one and two and five and six on the basis of the utility of gambling model explained above and on the basis of an "actuarial" model, that is, the model consisting of the hypothesis that Ss simply choo$e options so as to maximize expected monetary outcome. The predictive results for the data of these sessions were classified into 9 categories: (1) correct (winning) predictions for both actuarial and -7-

utility of gambling models (WW); (2) correct prediction for actuarial model, tie for utility of gambling (WT); (3) correct prediction for actuarial model, wrong (losing) prediction for utility of gambling (WL); (4) tie for actuarial model, win for gambling (TW); (5) tie for both models (TT); (6) tie for actuarial model, wrong prediction for gambling (TL); (7) wrong prediction for actuarial, right for gambling (LW); (8) wrong prediction for actuarial, tie for gambling (LT); (9) wrong prediction for both models (LL). The summary data comparing the actuarial and utility of gambling models for both sets of

~s, sailors and Stanford students, are presented in Table I, sailors first

and students after. (Data for three of the students were omitted because these Ss' choices were consistently actuarial.) Following Table I are two typical utility of gambling surfaces, those of Ss # I and 4t' 9.

Since no interesting conclusions about the two models can be drawn from the gross results recorded in Table I, a more careful analysis of the data

seems to be suggested. The question we are essentially concerned with is whether one of the models has greater predictive worth than the other.

That is, we wish to test the null hypothesis that the two models (gambling

and actuarial) are the same in prediciive power. To do this we first tabulate for each S the disagreements in predictions for the two models, then find, for each ~, the significance level for rejecting the null hypothesis, and finally we combine these significance levels and apply 2 a X test. References for this type of statistical analysis are Mosteller and Bush [3] and Moses [2]. -8-

TABLE I

Predictive Comparison of Gambling and Actuarial Models

Subject WW WT WL TW TT TL LW LT LL Totals

1 45 3 9 12 0 3 17 3 20 112 2 39 4 17 3 1 II 7 3 27 112 3 53 3 8 II 2 2 16 0 17 112 4 37 2 13 8 1 6 16 1 28 112 5 44 2 17 8 1 6 13 1 20 112 6 47 2 13 13 0 2 22 1 12 112 7 47 1 10 10 0 5 24 0 15 112 8 35 1 18 4 2 9 14 3 26 112 9 82 2 16 7 0 5 14 2 22 150 10 77 2 19 7 0 5 29 0 II 150 II 96 7 12 7 2 2 6 1 17 150 12 91 3 18 7 0 5 3 0 23 150 13 98 2 9 5 0 7 7 0 22 150 lI'otals 791 34 179 102 9 68 188 15 260 1646 -9-

Fig~re l. Utility of Gambling Surface for SUbject *Fl

80

+'22 + '2.0, +\5 +10 + 14

+- \'2 +\0 +B~ +2 +10 t4

+'2. o ° o- -20 -10 o 10 20 30 MeANS -10-

FigurE' 2. Utility of Gambling Surface for Subject #9

-4

~} -2 30 L -2

20 If) 1U 0 lJ z III ~ fO IJ.I u. I!- 0

-30 -20 _10 0 10 '2.0 -11-

The predictive disagreements were recorded in the f'ol,lowing way. Of'f'ers f'or which both models made the same prediction were ignored. Of'f'ers f'or which the utility of' gambling model predicted a correct choice, and the actuarial model predicted an incorrect choice or tie were given a plus (+). Any of'f'er f'or which the gambling model predicted a tie and the actuarial model a loss was also asSigned a pl~s. Of'f'ers f'or which, in the above description, the roles of' the gambling and actuarial models are reversed were assigned a minus (-). The data are recorded in Table II, the eight sailors f'irst, then the f'iVe students. Let p be the probability of' a plus. The exact signif'i- cance level was f'ound f'or the null hypothesis that p = .5 against the alterna- tive that p > .5. Since a two-sided test would not properly account for the varying directionality of' results, a one~sided test was used to provide a clear basis for combining significance levels. A similar one-sided test was made against the alternative that p < .5.

The 13 signif'icance levels (for both sailors and stUdents) were then combined, using, following the suggestion of' Moses t2], the L.ancaster COrreC- tion f'or continuity

13 p(X.) + p(X. + 1) (l) :z.:= -2 109 (~ ~ ) i=l 2 for Fisher's statistic

13 (2) t=: -2 log p(X ) i=l i where p(Xi ) is the significance level and Xi the observed number of pluses -12-

TABLE II

Pred~ct~ve D~sagreement of Gambling and Actuarial Models

Plus Minus Number 8)lbject (Gambling) (Actuarial) of Runs 1 31 15 26 2 12 32 18 3 28 12 18 4 25 21 22 5 22 25 23 6 36 17 26 7 34 16 18 8 22 28 27 9 23 23 30 10 36 26 32

11 14 2l 24 12 10 26 18 13 12 18 18 -13-

th 2 for the i subject. Statistic (1) has approximately the X distribu- tion with 26 degrees of freedom, two for each component. 2 2 The X results for the 13 Ss together were X = 47.180 against 2 the hypothesis that p> ·5, and, X = 40.406 against the hypothesis that p < .5. The first value is significant, for a two-sided test at the .01 level, the second, significant at the .05 level. Both tests resulted in a significance level ~ .05. Thus it seems advisable to partition Ss into the two SUbgroups into which they naturally fall (sailors and stUdents) and 2 apply the X test to each subgroup separately.

The X~6 value for the sailors, against the hypothesis that p > .5 is 40.630, which is significant for a two-sid,ed test at the .002 level. The

X~O value for the Stanford students against the s~e hypothesis is 6.55, which gives a significance level greater than .99. Testing the groups 2 individually against the hypothesis that p < .5, the X values were

19.504 for the sailors and 20.902 for the students, with respective signi- ficance levels for a two-sided test of .5 and .05. The clear-cut character of the statistical results for the sailors and students suggests that the development of a well-defined utility of gambling varies considerably in different cultural groups.

It might be objected that Ss' choices between the given pairs of offers were probably not independent, and hence that the above-described statistical analysis is not applicable. However, since almost all of the offers, with the exception of seven repetitions for the sailors and ten repetitions for the students in the first two sessions, were different, there should have been no strong negative or positive recency effects. One partial -14-

check of independence is given by the following test on the number of runs

of +'s and -'s. The basic idea is this, that if S has used some

systematic method of making choices, then the number of runs recorded

should be less than that expected on a random basis. For each SUbject

we tested the null hypothesis that the +'s and -'s are randomly distri-

buted against the alternative that there are too few runs. The obserVed

number of runs for each subject is recorded in Table II. The significance

levels obtained were combined using statistic (1) as explained above. The

result was X~6 = 14.309, which for a one-sided test becomes significant at

the .97 level. Thus our hypothesis of independence is well su.pported.

For another test of the utility of gambling model we may compare its

predictive worth against that of the non-linear utility model presented in

Suppes and Walsh [4]. The first eight §.s, the sailors, were used in both

experiments. Predictive disagreements between the two models were treated

in the same manner as predictive disagreements between the actuarial and

utility of gambling models described above.!! The individual significance

levels were found for each subject for the null hypothesis p = .5 first

against the hypothesis that p > .5 and second against the hypothesis that

p < .5. Statistic (1) was again used to combine the individual levels. The xi6 value against the first alternative, that p> .5, was 53.922, which is significant (for a two-sided test) at beyond the .001 level. 2 Against the alternative that P < .5, ~6 was 27.25 which is significant I I Pluses were assigned to correct predictions of the utility model) minuses I to correct predictions of the gambling model. f -15-

at .05. Thus, although both significance levels are <.05, the results favor the non-linear utility model.

Finally, there are some qualitative remarks which seem worth making about the utility of gambling surfaces.

(i) Only two ~s, both students, had a surface which was everywhere positive (for the range of options tested). The fact that none of the eight sailors had such a surface is evidence against the popular conception that service men will "gamble on anything." (Yet one sailor was indifferent between the options (-2i, -2i) and (+4-i, - ¢5.00).') No ~, sailor or student, had a surface which was always negative. These results suggest that any simple categorizatiOn of Ss into the two classes: having a taste for gambling, having a distaste for gambling, is bound to be grossly distorting.

(ii) Six sailors and three students had a high positive utility of gambling for options with large negative means and medium standard devia­ tions. Four sailors and two students had a negative utility of gambling for options with approximately zero means and large standard deviations.

Seven sailors and two students had noticeably stronger gradients in the area defined by options with negative means than in the corresponding positive area. The existence of these quasi-uniformities of behavior should be useful in constructing a more complicated and sophisticated theory of choice behavior.

(iii) In constructing the individual utility of gambling surfaces (each based on 27 observations), it was often difficult to draw contour lines adequately accounting for each observation. It should be interesting to -16-

investigate what kind of results of the sort described in the preceding

'paragraph would be obtain.ed on the basis of a very large number of observa­

tions with the contour lines drawn to fit "local" mean values of the observa-

tions.

Summary

This study reports experimental results for a utility of gambling model.

The model is designed to yield a direct measurement of utility of gambling.

The behavioral postulate of the model is that Ss choose among options so

as to maximize the sum of the expected monetary value and the utility of

gambling.

The Ss were eight sailors from Moffett Field Air Base and eight

undergraduate students at Stanford University. Each S participated in

six sessions, which primarily consisted of choosing between pairs of options

having small gains or losses of money as outcomes, depending on the throw

of a die. For each S 27 subsets of offers yielded 27 observations used

to construct a utility of gamblin.g surface.

The predictive wOrth of the utility of gambling model was compared with

the actuarial moq.el ()naXimization of expected money value) and found to be

Significantly better for the sailors and significantly worse for students, 2 on the basis ofax test which combined individual significance levels. -17-

REFERENCES

[1] D. Davidson, P. Suppes and S. Siegal. Decision Making: An Experimental Approach, Stanford University Press, Stanford, California, 1957.

[2] L. E. Moses, "Statistical theory and research design," AnnUal Review of , Yolo 7 (1956), pp. 233-258.

[3] F. Mosteller and R. R. Bush, "Selected Quantitative Techniques," Chapter 8, Handbook of Social Psychology, edited by G. Lindzey, Addison-Wesley Publishing Co., Cambridge, Massachusetts, 1954.

[4] P. Suppes and K. Walsh, "A non-linea.r model for the experimental measure­ ment of utility," Technical Report No. 11 (0l'lR contract l'lR 171-034), Stanford University, August, 1957.

[5] ;J. von Neumann and O. Morgenstern, Theory .9! Games and Economic Behavior, 2nd ed., Princeton University Pre.ss, Princeton, 1947. STAEFORD UNIVERSITY Technical Reports Distribution List Contract Nonr 225(17) (NR 171-034)

ASTIA Documents Service Center Office of Naval Research Knott Building Mathematics Division, Cbde 430 Dayton 2, Ohio 5 Department of the Navy Washington 25, D. c. 1 Commanding Officer Office of Naval Research Operations Research Office Branch Office 7100 Connecticut Avenue Navy No. 100, Fleet Post Office Chevy Chase, Maryland New York, New York 35 Attn: The Library 1

Director, Naval Research Office of Technical Services Laboratory Department of Commerce Attn: Technical Information Washingtdn 25, D. C. 1 Officer Washington 25, D. c. 6 The Logistics Research Project The George Washington University Office of Naval Research 707 - 22nd Street, N. W. Group Psychology Branch Washington 7, D. C. 1 Code 452 Department of the Navy The RAND Corporation Washington 25, D. C. 5 1700 Main Street Santa Monica, Calif. Office of Naval Research Attn: Dr. John Kennedy 1 Branch Office 346 Broadway Library New York 13, New York Cowles Foundation for Research in Economics Office of Naval Research Box 2125 Branch Office Yale Station 1000 Geary Street New Haven, Connecticut 1 San Francisco 9, Calif. Center for Philosophy of Office of Naval Research Science Branch Office University of Minnesota 1030 Green Street MinneapOlis 14, Minnesota 1 Pasadena 1, California 1 Professor Maurice Allais Office of Naval Research 15 Rue des Gates-Ceps Bral;1ch Office Saint-Cloud, (S.-O.) Tenth Floor France 1 The John Crerar Library Buildil;1g 86 East Randolph Street Professor K. J. Arrow Chicago 1, Illinois 1 Department of Economics Stanford University Office of Naval Research Stanford, California 1 Logistics Branch, Code 436 Department of the Navy Washington 25, D. C. 1 -ii-

Dr. R. F. Bales Dr. Francis J. Di Vesta Department of Social Relations Department of Psychology Syracuse University Cambridge, Massachusetts 1 123 College Place Syracuse, New York 1 Dr. Alex Bavelas Bell Telephone Laboratories Professor Robert Dorfman Murray Hill, New Jersey 1 Department of Economics Harvard University Professor Gustav Bergman Cambridge 38, Massachusetts 1 Department of Philosophy State University of Iowa Dr. Ward Edwards Iowa City, Iowa 1 Lackland Field Unit No.1 Operator Laboratory Professor E. W. Beth Air Force Personnel &Training Bern, Zweerskade 23, I Research Center Ams terdam, Z., San Antonio, Texas 1 The Netherlands 1 Professor W. K. Estes Professor R. B. Braithwaite Department of Psychology King's College Indiana University Cambridge, England 1 Bloomington, Indiana 1

Professor C. J. Burke Professor Robert Fagot Department of Psychology Department of Psychology Indiana University University of Oregon Bloomington, Indiana 1 Eugene, Oregon 1

Professor R. R. Bush Dr. Leon Festinger The New York School of Social Department of Psychology Work Stanford University 1 2 East Ninety-first Street Professor M. Flood New York 28, New York 1 Willow Run Laboratories Ypsilanti, Michigan 1 Dr. Donald Campbell Department of Psychology Professor Maurice FrechetI Northwestern University Institut H. Poinca:!e Evanston, Illinois 1 11Rue P. Curie Pads 5, France 1 Dr. Clyde H. Coombs Department of PsycholOgy Dr. Murray Gerstenhaber University of Michigan University of Ann Arbor, Michigan 1 , Pennsylvania

Dr. Mort Deutsch Dr. Leo A. Goodman Graduate School of Arts Statistical Research Center &Sciences University of Chicago New York University Chicago 37, Illinois 1 Washington Square New York 3, New York 1 -11i-

Professor Harold GullikSen Professor T. C. Koopmans Educational Testing Service Cowles Foundation for Research 20 Nassau Street in EconomicS Princeton, New Jersey 1 Box 2125, Yale Station New Haven, Connecticut 1 Professor Louis Guttman Israel Institute of Applied Professor Douglas Lawrence Social Research Department of Psychology David Hamlech No.1 Stanford University 1 Jerusalem, Israel 1 Dr. Duncan Luce Dr. T. T. ten Have Department of Social Relations Social - Paed.• Institmrt Harvard University Singel 453 Cambridge 38, Massachusetts 1 Amsterdam, Netherlands 1 Dr. Nathan Maccoby Dr. Harry Helson Boston University Graduate Department of Psychology School University of Texas Boston 15, Massachusetts 1 Austin, Texas 1 Professor Jacob Marschak Professor Carl G. Hempel Box 2125 Yale Station Department of Philosophy New Haven, Connecticut l Princeton University Princeton, New Jersey 1 Professor G. A. Miller Department Of Psychology Dr. Ian P. Howard Harvard UniverSity Department of Psychology Cambridge 38, MassachuSetts 1 University of Durham 7, KepierTerrace Dr. O. K. Moore Gilesgate Department of Sociology Durham, England 1 Box 1965 Yale Station Professor Lyle V. Jones New Haven, Connecticut 1 Department of Psychology University of North Carolina Professor Oskar Morgenstern Chapel Hill, North Carolina 1 Department of Economics &Social Institutions Dr. William E. Kappauf Princeton University Department of Psychology Princeton, New Jersey 1 University of Illinois Urbana, Illinois 1 professor Frederick Mosteller Department of Social Relations Dr. Leo Katz Harvard University Department of MathematicS Cambridge 38, Massachusetts 1 Michigan State College East Lansing, Michigan 1 Dr. Thillodore M. Newcomb Department of Psychology University of Michigan Ann Arbor, Michigan 1 -iv-

Dr. Helen Peak Dr. Sidney Siegel Department of Psychology Center for Behavioral Sciences University of Michigan 202 Junipero Serra Blvd. Ann Arbor, Michigan 1 Stanford, California 1

Professor Nicholos Rashevsky Professor Herbert Simon University of Chicago Carnegie Institute of TeChnology Chicago 37, Illinois 1 Schenley Pli\rk Pittsburgh, Pennsylvania 1 Dr. Frank Restle Department of Psychology Dr. Herbert Solomon Michigan State University Teachers College East j:.ansing, Michigan 1 Columbia University New York, New York 1 Professor David Rosenblatt American University Professor K.W. Spence Washington GJ D. C. 1 Psyc~ology Department State University of Iowa Professor Alan J. Rowe Iowa City, Iowa 1 Management Sciences Research Projeet Dr. F. F. Stephan University of California Box 337 Los Angeles 24, California 1 Princeton University Princeton, New Jersey 1 Dr. George Saslow Department of Psychiatry Dr~ Dewey B. Stuit University of Oregon Medical 108 Schaeffer Hall School State University of Iowa Portland, Oregon 1 Iowa City, Iowa I

Professor L. J. Savage Professor Alfred Tarski Committee on Statistics Department of Mathematics University of Chicago University of California Chicago, Illinois Berkeley 4, California 1

Dr. C. P. Seitz Dr. Robert L. Thorndike Special Devices Center Teachers College Office of Naval Research Columbia University Sands Point New York, New York 1 Port Washington Long Island, New York 1 Professor R. M. Thrall University of Michigan Dr. Marvin Shaw Engineering Research Institute School of Industrial Management Ann Arbor, Michigan 1 Massachusetts Institute of Technology pr. Masanao Toda 50 Memorial Drive Department of Experimental Cambridge 39, Massac)l.Usetts 1 Psychology Faculty of Letters Hokkaido University Sapporo, Hokkaido, Japl>n 1 -v-

Dr. E. Paul Torrance Professor Herman Rubin Survival Research Field Unit Department of Mathematics Crew Research Laboratory University of Oregon AFP & TRC Eugene, Oregon 1 Stead Air Force Base Reno, Nevada 1 Professor Ernest Adams Department of Philosophy Professor A. W. Tucker University of California Department of Mathematics Berkeley 4, California 1 Princeton University, Fine Hall Princeton, New Jersey 1 Professor Richard C. Atkinson Department of Psychology Dr. Ledyard R. Tucker I University of California Educational Testing Service Los Angeles 24, California 1 20 Nassau Street Princeton, New Jersey 1 Dr. David La Berge Department of Psychology Professor Wdward L. Walker University of Indiana Department of Psychology Bloomington, Indiana 1 University of Michigan Ann Arbor, Michigan 1 Dr. Samuel Messick Educational Testing Service Professor Tsunehiko Watanabe Princeton University Economics Department Princeton, New Jersey 1 Stanford University Stanford, California 1 Professor Norman H. Anderson Department of Psychology Dr. John T. Wilson Yale University National Science Foundation 333 Cedar Street 1520 H Street, N. W. New Haven, Connecticut 1 Washington 25, D. C. 1 Mr. Gordon Bower Professor J. Wolfowitz Department of PsychOlogy Department of Mathematics Yale University Cornell University New Haven, Connecticut 1 Ithaca, New York 1 Dr. Jean Engler Professor O. L.Zangwill Institute of Statistics Psychology Laboratory University of North Carolina Downing Place Chapel Hill, North Carolina· 1 Cambridge, England 1 Mr. Saul Sternberg Dr. I. Richard Savage Department of Social Relations School of Business Emerson Hall University of Minnesota Harvard University Minneapolis, Minn. 1 Cambridge 38, Mass. 1 -vi-

Professor Kellog Wilson Professor Max Black Department of Psychology Department of Philosophy Duke University Cornell University Durh8JIl-, NOrth Carolina 1 Ithaca, New York 1

Dr. Donald W. Stilson Professor C. West Churchman Department of Psychology Department of Industrial Engineering University of Colorado Case Institute of Technology Boulder, Colorado 1 University Circle Cleveland 6, Ohio 1 Professor T. W. Anderson Center for Behavioral Sciences Professor Sidney Morgenbesser 202 Junipero Serra Blvd. Department of Philosophy Stanford, California 1 Columbia University New York 27, New York 1 Professor Ernest Nagel Department of Philosophy Professor John G. Kemeny Columbia University Department of Mathematics New York 27, New York 1 Dartmouth College Hanover, New Hampshire 1 Professor Leonid Hurwicz School of Business Professor Nelson Goodman University of ~innesota Department of Philosophy Minneapolis 14, ~inn. 1 University of Pennsylvania Philadelphia, Pa. 1 Professor David Blackwell Department of Statistics Professor Howard Raiffa University of California Department of Statistics Berkeley 4, California 1 Harvard University Cambridge 38, Mass. 1 Dr. Eugene Galanter Department of Psychology Professor Willard V. Quine University of Pennsylvania Department of Philosophy Philadelphia 4, Pa. 1 Emerson Hall Harvard University ProfessorG. L. Thompson Cambridge 38, Mass. 1 Department of Mathematics Dartmouth College Mr. Dana Scott Hanover, New Hampshire 1 Department of Mathematics princeton University Dr. Hilary Putnam Princeton, New Jersey 1 Department of Philosophy Princeton University Professor Rudolf Carnap Princeton, New Jersey 1 Dep~rtment of Philosophy U.C.L.A. Professor S. Smith Stevens Los Angeles 24, California 1 Memorial Hall Harvard University Cambridge 38, Mass. 1 -vii-

Professor Donald Kalish Department of Philosophy University of California Los Angeles 24, California 1

Professor Morton White Department of Philosophy Harvard University Cambridge 38, Mass. 1

Additional copies fOr project leader and assistants and re- serve for future requirements 25